The Sun claim Fleetwood Town and Accrington Stanley are eying the Burnley youngster, who will be allowed to leave on loan this term.
Dunne, a Republic of Ireland under-21 international, spent the second half of last season at the Stadium of Light, making 14 appearances in all competitions and netting one goal.
In other news, Cats youngster Bali Mumba has been named in the England under-19 side for their upcoming fixtures – the 17-year-old, who can play in midfield or at right-back, made his senior debut in Sunderland’s win against Wolves on the last day of the 2017-18 season.
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Paul Simpson's side will begin their series of fixtures at St. George’s Park on Thursday 5 September when Greece are the visitors to Burton, before the squad head to Germany for a second fixture on Monday 9 September.
Elsewhere in League One, Blackpool are reportedly close to signing winger Sean Scannell – the 28-year-old is said to be in ‘advanced talks’ at Bloomfield Road.
Ex-Sunderland and current Tangerines boss Simon Grayson is a fan Scannell having worked with the versatile attacker during his time in charge of Huddersfield Town.
Meanwhile, Bolton’s caretaker manager Jimmy Phillips wants to sign four players before Saturday's game at Gillingham with former defender Jack Hobbs, 31, having signed a deal to re-join the club.
Finally, Bury are "actively considering" legal action against the English Football League following the club's "unfair" expulsion on Tuesday.